A relation on a linearly ordered structure is called
semi-bounded if it is definable in an expansion of
the structure by bounded relations.
We study ultimate behavior of semi-bounded relations
in an ordered module M over an ordered commutative ring R
such that M/r M is finite for all nonzero r∈ R.
We consider M as a structure in the language of ordered
R-modules augmented by relation symbols for the submodules rM,
and prove several quantifier elimination results for semi-bounded
relations and functions in M. We show that these quantifier
elimination results essentially characterize the ordered
modules M with finite indices of the submodules rM.
It is proven that
(1) any semi-bounded k-ary relation on M
is equal,
outside a finite union of k-strips, to
a k-ary relation quantifier-free definable in M,
(2) any semi-bounded function from Mk to M
is equal,
outside a finite union of k-strips, to
a piecewise linear function, and
(3) any semi-bounded in M
endomorphism of the additive group of M
is of the form x ↦ σ x, for some σ from
the field of fractions of R.